## Abstract

The vibration and sound radiation characteristics of a double leaf panel wall with corrugated core are analyzed. The core comprises of repeating unit cells along the length of the panel. Each cell is a combination of a straight beam with uniform cross section and curved beams with varying cross section. The vibroacoustic property of the corrugated panel is analyzed assuming plain strain condition. The analysis presented in the work shows that the proposed sandwich panel design provides broadband sound transmission loss (STL) characteristics. The dispersion analysis, forced response characteristics, and sound radiation characteristics of the structure are presented. It is found that bending frequency band gap obtained from the dispersion analysis is a prominent criterion to achieve a higher STL. The structural and acoustic behaviors of the sandwich panel are analyzed through the spectral finite element model. The STL for the core element for various taper ratios is presented. It is found that a higher taper ratio shifts the STL characteristics to the low frequency range.

## 1 Introduction

Cellular solids have shown great potential in making lightweight thin structures [1], which have application and demand in aerospace, automobile, and real estate. They are being used as a core material in a number of applications such as vibration control [2,3], sound absorption [4], and heat dissipation attribute [5]. In most commercial applications of lightweight panel walls, the core material comprises of foam and cellular solids [6]. The microstructures of foams are randomly oriented and its stochastic nature is responsible for its sound absorbing nature. The interaction of the air and the porous structure of foam is the primary cause of high damping at higher excitation frequencies. The core with a deterministic structure is studied in the past for obtaining its sound transmission loss (STL) [7]. Analytical and theoretical models were developed to predict the sound transmission loss in an isotropic and orthotropic cores [8,9]. In this respect, a periodic truss core panel has been studied by El-Raheb and Wagner [10]. The effect of fluid loading and damping on the sound panel transmission characteristics is studied for varying panel aspect ratio [11]. Ruzzene [7] extended the work with honeycomb core and effects of core geometry on the structural response and acoustic radiation are communicated. To minimize the transmitted sound power, topology optimization techniques were used to design the core [12,13].

The study on the sandwich plate has mostly remained confined to core elements comprising of straight beams having constant area of cross section. The sound transmission characteristics of sandwich double leaf panel having core, comprised of curved beam element with tapered cross section is not reported in the literature so far. In this paper, we present a unified approach that takes advantage of the free wave propagation phenomena of unit cell and its effect on sound transmission characteristics in the double leaf sound panel with tapered curved core (DLPTC). The tapered curved core and the term “corrugated” are used interchangeably. First, we develop the spectral element of the curved and tapered elements. The DLPTC is designed based on the unit cell comprising of the core and the face plate. The free wave vibration analysis of the infinite structure is performed using transfer matrix in conjunction with Floquet–Bloch theorem considering a single-unit cell. The dynamics attribute of the finite DLPTC is then computed using a spectral finite element model. The transmitted sound in the fluid domain is then evaluated using the Rayleigh integral with displacement field as the input. In this work, the fluid domain is assumed to be much less dense than the structure, such as air. As the structure is coupled to a light fluid, it is considered as weak coupling [14]. Thus, the vibration response of the structure is obtained neglecting the fluid coupling. Also, the acoustic response is obtained by using the vibration of structure as boundary condition.

## 2 Theoretical Formulation

The schematic diagram of the DLPTC is shown in Fig. 1. It consists of two face plates and a tapered curved core. The DLPTC is assumed to be infinite in *z*-direction. There is no variation of strain in the *z*-direction. As a result, the three-dimensional problem can be treated as a plain strain problem and the DLPTC is represented with a geometry, shown in Fig. 1(b).

In line with the conventional method of modeling the sandwich panel with cellular cores [7,10,12], the face plate and the core are considered as frame elements. The members of the core are modeled as curved and tapered beams. The taper ratio is defined as the ratio of the thickness of the beams at two ends is *h*_{1}/*h*_{2}. The frame element has slenderness ratio given by *h*_{2}/*L* = 0.2 and bent angle, *ϕ* = *L*/*R*, where *L* is the length of the curved beam and *R* is the radius of curvature. Thus, a specific value of *θ*, taper ratio *h*_{1}/*h*_{2}, and slenderness ratio defines a particular configuration of the square DLPTC, as shown in Fig. 1(b). The top face plate faces the unbounded fluid domain; however, the bottom face plate is subjected to a normally incident uniform harmonically varying pressure wave.

## 3 Formulation of the Dynamic Stiffness Matrix

### 3.1 Dynamic Stiffness Matrix of Curved and Tapered Beams.

*c*

_{1}and

*c*

_{2}are conicity associated with beams 1 and 2. They are defined as

*c*

_{1}= (

*h*

_{2}−

*h*

_{1})/

*h*

_{1},

*c*

_{2}= (

*h*

_{2}−

*h*

_{1})/

*h*

_{2}, where

*h*

_{1}and

*h*

_{2}represent the thickness of the beam 1 at

*s*= 0 and

*s*=

*L*, respectively. Using the conicity, the cross-sectional area and moment of inertia of beams 1 and 2 are given by

*A*

_{0j}and

*I*

_{0j}are the area and moment of inertia of the cross section at the arc length coordinate

*s*

_{j}= 0, where

*j*= 1, 2. Without the loss of generality the force-displacement relation in the beams 1 and 2 are given as [15]

*M*is the moment of inertia,

*F*is the normal force,

*V*is the shear force across the cross section at a particular arc length

*s*.

*E*is the Young’s modulus,

*ρ*is the density,

*w*and

*u*represent the transverse and longitudinal displacements of the centroidal points.

*ψ*= (

*u*/

*R*) − d

*w*/d

*s*represents the rotation of the cross section.

*s*of the right-hand side of Eq. (3) is omitted for brevity. Following the procedure discussed by Lee [16], the state-space equation beam 1 can be written as follows:

_{1}is given by

*g*

_{1}=

*g*(

*s*) = (1 +

*c*

_{1}

*s*/

*L*).

Note, the matrix B_{1} is a function of the arc length coordinate *s*. As such, Eq. (4) cannot be directly integrated to derive the transfer matrix. Thus, to derive the transfer matrix (*C*_{1}) for beam 1, we implement the Floquet theory [17].

_{2}is given by

*g*

_{2}=

*g*

_{2}(

*s*) = (1 −

*c*

_{2}

*s*/

*L*). Similar to beam 1, we derive the transfer matrix (

*C*

_{2}) for beam 2 using the procedure detailed in Ref. [17].

Note, the transfer matrices for beams 1 and 2 are derived in their respective coordinate systems. The transfer matrix of each beam in the unit cell is valid in the local coordinate system of the respective beams. Therefore, we first transform the transfer matrices from local coordinate system to global coordinate system before applying the continuity conditions. A detailed discussion about the transformation can be found in Refs. [3,18]. Using the procedure discussed in Ref. [16], we obtain the global dynamic stiffness matrix from the global transfer matrix. Let $D2,D3,andD4$ be the dynamic stiffness matrices of the curved beams between nodes (4, 5), (2, 5), and (5, 6), respectively (refer to Fig. 1(c)).

### 3.2 Dynamic Stiffness of the Face Plate.

*η*is the axial coordinate,

*U*(

*η*,

*t*),

*W*(

*η*,

*t*) is the axial and transverse displacements in the local coordinate system.

*I*and

*A*are the moment of inertia and area of cross section of the straight beam. Considering a harmonic solution in the form

*U*=

*u*(

*η*)

*e*

^{iωt}and

*W*=

*w*(

*η*)

*e*

^{iωt}, where

*ω*is the free wave frequency, the axial displacement u(

*η*), and transverse displacement

*w*(

*η*) is written in spectral form as

*F*, shear force

*V*(

*η*), and bending moment

*M*(

*η*) in the frame is given as

*B*= {

*B*

_{1},

*B*

_{2},

*B*

_{3},

*B*

_{4},

*B*

_{5},

*B*

_{6}} is given as

*B*} from Eq. (10) is used in Eq. (12) to obtain

*D*

_{e}is the elemental dynamic stiffness in the local coordinate system. Let

*D*

_{1}and

*D*

_{5}be the dynamic stiffness matrices (in the global coordinate system) of the straight beams between nodes (1, 4) and (6, 3), respectively (refer to Fig. 1(c)).

## 4 Matrix Equation for Axial and Flexural Displacements

*N*

_{i}(

*η*,

*ω*) for i=1,6, are the shape functions. The shape function matrix, which is a continuous function of the axial coordinate (

*η*), along with the nodal displacement vector can be used to obtain the displacement of a point anywhere in the frame element.

*P*

_{i}(

*η*,

*ω*) =

*P*

_{i}

*e*

^{iωt}, where

*θ*is the angle of incidence of the acoustic pressure when calculated with respect to

*y*-axis,

*k*=

*ω*/

*c*

_{air}is the wave number of the acoustic pressure wave, and

*c*

_{air}is the speed of the sound in air.

### 4.1 Nodal Load Due to Incident Pressure.

*P*. The equivalent nodal load over the element is obtained using the principle of virtual work [7]. The work done by the distributed load

*P*(

*η*,

*ω*) over the element is given as

*δw*is the virtual flexural displacement of the bottom face and

*b*is the width of the beam. The virtual flexural displacement

*δw*is interpolated in terms of shape functions [

*N*(

*η*,

*ω*] = {

*N*

_{3}(

*η*,

*ω*),

*N*

_{4}(

*η*,

*ω*),

*N*

_{5}(

*η*,

*ω*),

*N*

_{6}(

*η*,

*ω*)} taking arbitrary constants vector

*δd*as

*δw*= [

*N*(

*η*,

*ω*]

*δd*

## 5 Radiation Pressure in Acoustic Domain

*P*

_{t}(

*r*,

*θ*,

*ω*) in the acoustic fluid domain due to incident acoustic pressure

*P*

_{i}on the bottom face of the panel wall is given by Rayleigh’s integral [21] as

*k*=

*ω*/

*c*

_{air}is the wave number of the acoustic pressure wave,

*ρ*

_{a}is the density of the fluid domain, $R(x)=(rcos\theta \u2212x)2+(rsin\theta )2$ is the radial distance between a point on the beam element having velocity

*v*(

*x*) =

*iωw*and the observation point, $H01$ is the Hankel function of the first kind,

*L*

_{t}is the total length of the structure. The transmitted sound power over the receiver surface

*S** in the far field shown in Fig. 1(b) is given by

## 6 Numerical Verification

In this section, the numerical verification of the aforementioned methodology is presented. In order to verify the procedure, the transmission loss calculated by Griese et al. for the hexagonal sandwich beam is reproduced using the aforementioned methodology. It is to be noted the hexagonal core elements by Griese et al. [4] were comprised of straight frame elements with no curvature and taper. In the aforementioned methodology, straight beam for the core is obtained by substituting bent angle *ϕ* = 0 and conicity *c*_{1} = 0. The obtained transmission loss is shown in Fig. 2, also overlaid is the transmission loss calculated using the effective medium theory [21]. It is seen the results obtained are in good correlation with both the mass law, spectral element method and those obtained by Griese et al. [4].

## 7 Dispersion Curve of the DLPTC Unit Cell

*D*

_{1},

*D*

_{2},

*D*

_{3},

*D*

_{4},

*D*

_{5}is assembled to obtain the global dynamic stiffness matrix [20]. Thus we get

*p*

_{i}=

*u*

_{i},

*v*

_{i},

*θ*

_{i}and

*u*

_{i},

*v*

_{i},

*θ*

_{i}refer to axial displacement, transverse displacement, and slope at the

*i*th nodal points. Similarly, the

*q*

_{i}=

*F*

_{i},

*V*

_{i},

*M*

_{i}represents the axial force, shear force, and bending moments at the

*i*th nodal points for i=1 to 6. The above equation can be rewritten in terms of its left nodes and right nodes as

*P*

_{L}=

*p*

_{1},

*p*

_{2},

*p*

_{3},

*P*

_{R}=

*p*

_{4},

*p*

_{5},

*p*

_{6},

*Q*

_{L}=

*q*

_{1},

*q*

_{2},

*q*

_{3}, and

*Q*

_{R}=

*q*

_{4},

*q*

_{5},

*q*

_{6}. The above equation can be rearranged to give transfer matrix relating generalized displacements and forces at the left end to the right end as

*T*] is the transfer matrix. Using the Floquet–Bloch theorem on the state vectors at the left end and right end as

*λ*=

*e*

^{iμ}. In the above equation, the transfer matrix is an 18 × 18 matrix. The solution of the above equation at a particular frequency gives 18 eigenvalues and 18 eigenvectors. Each eigenvectors correspond to the free characteristic wave and the eigenvalues give its corresponding phase constant

*μ*. The phase constant

*μ*is found by taking −

*i*ln(

*λ*). The phase constant

*μ*is complex in nature whose real part of $R(\mu )$ gives the propagation constant and the imaginary part $I(\mu )$ gives the attenuation constant for each of the 18 characteristic waves. A non-zero attenuation constant $I(\mu )$ for a characteristic wave implies that wave undergoes spatial attenuation. The dispersion curve is obtained by plotting the values of propagation constant $R(\mu )$ and attenuation constant $I(\mu )$ with varying frequency.

## 8 Finite Structure Response and Sound Transmission Loss

### 8.1 Geometry and Material Properties.

The double leaf panel with corrugated core considered is 2 m long and 5 cm thick. The face plate, as well as the core are made up of aluminum (*E* = 70 × 10^{9} N/m^{2}, *ρ* = 2700 kg/m^{3}). The maximum thickness of any member of the core is *h*_{2} = 2.5 mm while top and bottom face plates is *h*_{2}/2 = 1.25 mm thick and *L* = 0.0125 mm as shown in Figs. 1(c) and 1(d). The bent angle of all the core members is taken as *ϕ* = 30 deg and taper ratio *h*_{1}/*h*_{2} is taken as variable. The out of plane thickness *b* for all the members is considered to be unity. The fluid medium on both sides is considered to be air having density of air *ρ*_{a} = 1.2 kg/m^{3} and speed of sound in air *c*_{air} = 343 m/s. In all the configurations, the panel is placed in a rigid baffle with simply supporting boundary condition.

### 8.2 Panel Response.

*ϕ*= 0 and conicity

*c*

_{1}= 0, this implies that all the members of the core element are straight and have a uniform thickness of

*h*

_{2}= 2.5 mm. In Fig. 3(b), the transverse displacement response of the top face plate is presented in terms of root-mean-square velocity

*v*

_{rms}. The

*v*

_{rms}is defined as

*v*

_{rms}is presented in dB using a reference velocity of 5 × 10

^{−8}m/s [7,10]. Alongside of it in Fig. 3(a), is the dispersion curve obtained using the method presented in Sec. 7. For the frequency interval 4140–7545 Hz, the bending wave mode does not propagate. The frequency range is shown in Fig. 3(a) with gray shade. The mode shape of a characteristic bending wave is calculated using the eigenvector of Eq. (29) and using Eq. (15) to get the intermediate displacement values. The $I(\mu )$ is positive for the bending mode, as a result of that these particular wave mode attenuates spatially as it propagates. It is important to note that the structure does not possess complete band gap, as the other characteristic waves propagate in the structure. However, the band exists in different characteristic modes, such as bending, shear, longitudinal, and rotational modes [22].

In Fig. 3(c), the sound transmission loss through the structure for a normally incident sound pressure field is shown. At low frequency region, there is no significant transmission loss as the bending wave mode propagates, whereas the STL in the bending band gap mode, is predominantly high. The sound transmission loss drops at natural frequency of the simply supported structure, however, the average sound transmission loss over the band gap region is on a higher side when compared to the frequency region where the bending wave mode propagates. In particular, an average 76 dB of STL is observed in the band gap corresponding to the bending mode as shown in Fig. 3(c).

### 8.3 Effect of Varying Taper on Sound Transmission Loss.

The parameter taper ratio *h*_{1}/*h*_{2} is varied to understand its effect on the band gap in bending mode. The evolution of bending mode band gap as the conicity parameter is varied is shown in Fig. 5. It is seen that with decrease in the taper ratio the bandwidth of the mode decreases and the band starts at lower frequency. The dispersion curve, root-mean-square velocity, and STL for taper ratios *h*_{1}/*h*_{2} = 0.5, 0.1 are shown in Figs. 6 and 7. For a smaller taper ratio *h*_{1}/*h*_{2}, the performance of the panel system is improved in the low frequency interval of 2500–4700 Hz. When compared to the prismatic straight core beams where the band value is 4140–7545 Hz, the tapered core with bent angles provides a significant improvement toward achieving higher STL performance at lower frequency. Thus the corrugated core offers a better acoustic performance and an opportunity to tune the frequency to obtain the desired sound transmission loss performance.

## 9 Conclusion

This paper studies the vibration and acoustic performance of the double leaf sound panel with tapered curved core. The double leaf corrugated core is analyzed assuming plain strain condition. The proposed structure presents a viable structure for reduction of sound over a wide frequency of ranges. The parameters influencing the bandwidth over which the sound transmission loss is increased are studied. The analysis is performed by changing the core element geometry through taper ratio and bent angle of the individual element. A spectral finite element model is used to find the vibration response of the structure which is then coupled with the Rayleigh integral to find the pressure field in the semi-infinite acoustic domain. In addition to the dispersion curve of the infinite structure is presented to gain additional insight into the phenomena leading to increased transmission loss.

In this study, we elucidate that the sound transmission is high in those frequencies where the bending mode of free characteristics wave propagates. The frequencies in which the bending wave mode do not propagate, the sound transmission is very low. Hence a core can be suitably designed by designing the structure in such a that the sound frequency falls in the range where the bending wave modes do not propagate. Thus, the band gap in bending mode is a favorable condition for increased sound transmission loss. It is also found that for the corrugated core element, reduction of taper ratio to lower values shift the band gap in bending mode to lower frequency. Thus taper ratio along with bent angle of the core can act as tuning parameter to achieve broadband STL characteristics.

## Acknowledgment

Author acknowledges SERB-NPDF Grant No. PDF/2021/000087 and Indo-Canada IC impact Grant No. DST/INT/CAN/P-03/2020 for supporting the research.

## Conflict of Interest

There are no conflicts of interest.

## Data Availability Statement

The datasets generated and supporting the findings of this article are obtainable from the corresponding author upon reasonable request.